3.568 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^{3/2}}{x^{11}} \, dx\)

Optimal. Leaf size=72 \[ \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{40 a^2 x^{10}}-\frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{8 a x^{10}} \]

[Out]

-((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(8*a*x^10) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(40*a^2*x^10)

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Rubi [A]  time = 0.0167922, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {1110} \[ \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{40 a^2 x^{10}}-\frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{8 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^11,x]

[Out]

-((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(8*a*x^10) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/(40*a^2*x^10)

Rule 1110

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*x^2
+ c*x^4)^(p + 1))/(4*a*d*(p + 1)*(2*p + 1)), x] - Simp[((d*x)^(m + 1)*(2*a + b*x^2)*(a + b*x^2 + c*x^4)^p)/(4*
a*d*(2*p + 1)), x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && EqQ[m + 4*p + 5,
0] && NeQ[p, -2^(-1)]

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{11}} \, dx &=-\frac{\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{8 a x^{10}}+\frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{40 a^2 x^{10}}\\ \end{align*}

Mathematica [A]  time = 0.0132023, size = 61, normalized size = 0.85 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (15 a^2 b x^2+4 a^3+20 a b^2 x^4+10 b^3 x^6\right )}{40 x^{10} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/x^11,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(4*a^3 + 15*a^2*b*x^2 + 20*a*b^2*x^4 + 10*b^3*x^6))/(40*x^10*(a + b*x^2))

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Maple [A]  time = 0.164, size = 58, normalized size = 0.8 \begin{align*} -{\frac{10\,{b}^{3}{x}^{6}+20\,a{x}^{4}{b}^{2}+15\,{a}^{2}b{x}^{2}+4\,{a}^{3}}{40\,{x}^{10} \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/x^11,x)

[Out]

-1/40*(10*b^3*x^6+20*a*b^2*x^4+15*a^2*b*x^2+4*a^3)*((b*x^2+a)^2)^(3/2)/x^10/(b*x^2+a)^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.45071, size = 85, normalized size = 1.18 \begin{align*} -\frac{10 \, b^{3} x^{6} + 20 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} + 4 \, a^{3}}{40 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/x^11,x, algorithm="fricas")

[Out]

-1/40*(10*b^3*x^6 + 20*a*b^2*x^4 + 15*a^2*b*x^2 + 4*a^3)/x^10

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{x^{11}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/x**11,x)

[Out]

Integral(((a + b*x**2)**2)**(3/2)/x**11, x)

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Giac [A]  time = 1.13632, size = 93, normalized size = 1.29 \begin{align*} -\frac{10 \, b^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + 20 \, a b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 15 \, a^{2} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 4 \, a^{3} \mathrm{sgn}\left (b x^{2} + a\right )}{40 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/x^11,x, algorithm="giac")

[Out]

-1/40*(10*b^3*x^6*sgn(b*x^2 + a) + 20*a*b^2*x^4*sgn(b*x^2 + a) + 15*a^2*b*x^2*sgn(b*x^2 + a) + 4*a^3*sgn(b*x^2
+ a))/x^10